# Existence of codimension 1 topological foliations

One of the many famous theorems proven by William Thurston is that a closed connected smooth manifold $$M$$ admits a codimension 1 smooth foliation if and only if $$\chi(M)=0$$:

W.P. Thurston, Existence of codimension-one foliations. Ann. of Math. (2) 104 (1976), no. 2, 249–268.

The "only if" part of this theorem, of course, is much simpler and older.

My question is: What happens in the topological category?

I convinced myself (although I did not write down a detailed proof) that if a closed connected topological manifold $$M$$ admits a topological codimension 1 foliation, then $$\chi(M)=0$$. (The proof imitates the "standard" smooth argument, using a map from $$M$$ to the symmetric product of $$M$$ with itself, instead of constructing a line bundle on $$M$$ in the smooth case.) The true question is if the hard part of Thurston's theorem holds in the topological category:

Question. Suppose that $$M$$ is a closed connected topological manifold with $$\chi(M)=0$$. Does $$M$$ admit a codimension 1 foliation?

I could not find anything on MathSciNet using the obvious algorithm.

• I will wear the down-vote on this question as a badge of honor. – Moishe Kohan Apr 5 at 20:46
• Hello Moishe. Thurston's construction has two steps: 1) build the foliation on the complement of "holes"; 2) extend it through the holes. Since the holes are contained in a small neighborhood of a finite union of loops, you can smoothen the manifold there, hence the step 2 would make no problem. Today, the step 1 falls to the general methods of Gromov's h-principle, e.g. the Eliashberg-Mishachev Holonomic Approximation Theorem; so the actual question is to extend these methods to the topological case. Good luck. – Gael Meigniez Apr 14 at 6:48
• Maybe a more interesting and more accessible question would be to build codimension-one foliations in the PL frame. – Gael Meigniez Apr 14 at 6:50